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An inverse scattering problem for a perturbed Hill's operator

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Abstract

We consider the inverse scattering problem for the operator L=−d2/dx2+p(x)+q(x), x ∈ R1. The perturbation potential q is expressed in terms of the periodic potential p and the scattering data. We also obtain identities for the eigenfunctions of the unperturbed Hill's operator L0=−d2/dx2+p(x).

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Literature cited

  1. E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Pt. 2, Oxford University Press (1958).

  2. N. E. Firsova, “The Riemann surface of a quasi-impulse and the scattering theory for a perturbed Hill's operator,” in: Mathematical Problems of the Theory of Wave Diffusion [in Russian], Vol. 7, Izd. LOMI, Leningrad (1974), pp. 51–62.

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Authors and Affiliations

  1. Leningrad State University, USSR

    N. E. Firsova

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  1. N. E. Firsova
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Additional information

Translated from Matematicheskie Zametki, Vol. 18, No. 6, pp. 831–843, December, 1975.

The author thanks B. S. Pavlov for posing the problem, and D. R. Yafaev for some useful remarks.

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Firsova, N.E. An inverse scattering problem for a perturbed Hill's operator. Mathematical Notes of the Academy of Sciences of the USSR 18, 1085–1091 (1975). https://doi.org/10.1007/BF01099986

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  • DOI: https://doi.org/10.1007/BF01099986

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